Vibration analysis of a flexible rotating disk with angular misalignment

The dynamic characteristics and responses of a flexible rotating disk are analyzed, when the disk has angular misalignment that is defined by the angle between the rotation and symmetry axes. Based on the von Karman strain theory and the Kirchhoff plate theory, three equations of motion are derived for the transverse, radial and tangential displacements when the disk has angular misalignment. The derived equations are fully coupled partial differential equations through the transverse, radial and tangential displacements. In particular, the equation of transverse motion is non-linear while the others are linear. After these partial differential equations and the associated boundary conditions are transformed into a weak form, the weak form is discretized to a non-linear matrix–vector equation by using the finite element method. The non-linear equation is linearized in the neighbourhood of a dynamic equilibrium position, and then the natural frequencies and mode shapes are computed. In addition, the dynamic time responses are obtained by applying the generalized-α method. The effects of angular misalignment on the natural frequencies, the mode shapes and the dynamic responses are investigated. The analysis shows that the angular misalignment causes the natural frequency split and the out-of-plane mode with only one nodal diameter and no nodal circle has the largest frequency split. It is also found that the angular misalignment yields the amplitude modulations in the transverse, radial and tangential dynamic responses.